The heroic forbidden-set conjecture for oriented forests and heroes

Let $H$ be a hero, meaning a tournament such that every tournament not containing $H$ has bounded dichromatic number, and let $F$ be an oriented forest. A set of digraphs is heroic if every digraph with none of its members as an induced subdigraph has bounded dichromatic number. The heroic forbidden-set conjecture. The set $\{\overleftrightarrow{K_2},H,F\}$ is heroic if and only if either $F$ is the disjoint union of oriented stars or $H$ is a transitive tournament.

This is presented as an analogue of the Gyárfás–Sumner conjecture for oriented graphs. The only-if direction was proved in the cited prior work, while the conjecture is stated to be widely open.